Non-Markovian

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

Because of the general difficulty encountered in generating reliable potentials energy surfaces and estimating reasonable friction kernels, it still remains an open question whether by analysis of experimental rate constants one can decide whether non-Markovian bath effects or other influences cause a particular solvent or pressure dependence of reaction rate coefficients in condensed phase. From that point of view, a purely... 

Bigot J-Y, Portella M T, Schoenlein R W, Bardeen C J, Migus A and Shank C V 1991 Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes Phys. Rev. Lett. 66 1138 1... 

The quantum theory of spectral collapse presented in Chapter 4 aims at even lower gas densities where the Stark or Zeeman multiplets of atomic spectra as well as the rotational structure of all the branches of absorption or Raman spectra are well resolved. The evolution of basic ideas of line broadening and interference (spectral exchange) is reviewed. Adiabatic and non-adiabatic spectral broadening are described in the frame of binary non-Markovian theory and compared with the impact approximation. The conditions for spectral collapse and subsequent narrowing of the spectra are analysed for the simplest examples, which model typical situations in atomic and molecular spectroscopy. Special attention is paid to collapse of the isotropic Raman spectrum. Quantum theory, based on first principles, attempts to predict the. /-dependence of the widths of the rotational component as well as the envelope of the unresolved and then collapsed spectrum (Fig. 0.4). 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

The theory of Section 1.8 is sometimes qualified as non-Markovian since it accounts for non-exponential angular momentum relaxation, unlike impact theory which is Markovian in this sense. However, it is not a unique non-Markovian generalization of impact theory. Not less known is a differential version of the theory... 

Although non-Markovian, the differential theory surely has Markovian asymptotics at sufficiently long times ... 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

It is commonly believed that K (t ) may be carried outside the integral without lack of accuracy if inequality (2.23) is satisfied. This is the same way that was used in Chapter 1 to obtain the non-Markovian differential equation... 

Markovian theory of orientational relaxation implies that it is exponential from the very beginning but actually Eq. (2.26) holds for t zj only. If any non-Markovian equations, either (2.24) or (2.25), are used instead, then the exponential asymptotic behaviour is preceded by a short dynamic stage which accounts for the inertial effects (at t < zj) and collisions (at t < Tc). 

In Markovian approximation (zj =0) this quantity approaches the famous Debye plateau shown in Fig. 2.3 whereas non-Markovian absorption coefficient (2.56) tends to 0 when ft) — 0 as it is in reality. This is an advantage of the Rocard formula that eliminates the discrepancy between theory and experiment by taking into account inertial effects. As is seen from Eq. (2.56) and the Hubbard relation (2.28)... 

The mutual correspondence of non-Markovian and Markovian (impact) approximations becomes clear, if the second derivative of K/(t) is considered. It varies differently within three time intervals with the following bounds xc < xj < Tj1 (Fig. 2.5). Orientational relaxation occurs in times Fj1. The gap near zero has a scale of xj. A parabolic vertex of extent xc and curvature I4 > 0 is inscribed into its acute end. The narrower the vertex, the larger is its curvature, thus, in the impact approximation (tc = 0) it is equal to 00. In reality xc =j= 0, and the... 

We will show below when and how the line interference and its special case, spectral exchange , appear in spectral doublets considered as an example of the simplest system. It will be done in the frame of conventional impact theory as well as in its modern non-Markovian generalization. Subsequently we will concentrate on the impact theory of rotational structure broadening and collapse with special attention to the shape of a narrowed Q-branch. 

Quantum theory of spectral collapse 4.3 Non-Markovian binary theory... 

In non-Markovian theory, the off-diagonal elements of G cannot be looked upon as transfer rates between two or more discrete eigenfrequen-cies of the system as they are functions of the continuous variable to. The transfer rates concept is only acceptable in the Markovian limit of the theory (t tc) when co-dependence is eliminated. To obtain this limit, we must first pass to differential formulation of non-Markovian theory and after that let t -> oo. In the literature there is complete unanimity on how the transition from integral to differential formalism can be carried out correctly. According to Eq. (4.28) the integrand in Eq. (4.26) may be written as... 

It is once again the non-Markovian equation in a sense that the relaxation rates are time-dependent. They become constant for the times which are long enough to extend the integration over t to 00. This leads... 

Fig. 4.5. The broadening of the P-R doublet (Atc = n/2, V2f = n/8) in the integral non-Markovian theory (solid line) and in the Markovian approximation (dotted line).

NMR see nuclear magnetic resonance non-Markovian binary theory 138-40 non-Markovian differential theory 38-45, 65... 

For the interbipolycondensation the condition of quasiideality is the independence of the functional groups either in the intercomponent or in both comonomers. In the first case the sequence distribution in macromolecules will be described by the Bernoulli statistics [64] whereas, in the second case, the distribution will be characterized by a Markov chain. The latter result, as well as the parameters of the above mentioned chain, were firstly obtained within the framework of the simplified kinetic model [64] and later for its complete version [59]. If all three monomers involved in interbipolycondensation have dependent groups then, under a nonequilibrium regime, non-Markovian copolymers are known to form. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

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